Lemma 1

If B and C are lower triangular matrices of order n with diagonals $(\beta_i)$ and $(\gamma_i)$, then the matrices $B + C$ and $BC$ are lower triangular with diagonals $(\beta_i + \gamma_i)$ and $(\beta_i \gamma_i)$.

Since the matrix $U$ of $u$ is lower triangular with diagonal $(\alpha_i)$ say, it follows from Lemma 1 that $q(U)$ is a lower triangular matrix with diagonal $(q(\alpha_i))$. Then $X . I, -q(U)$ is a lower triangular matrix with diagonal $(X - q(a;))$, which proves (6).